Wondrous Mathematical Thinking

“In the past, mathematicians always tried to solve problems exactly. Now we realize that most problems will never have an exact solution. Nonetheless, we can hope to understand the general shape of a solution, and topology gives a language for talking about these shapes. Topology gives a new point of view on all kinds of physical phenomena: the collapse of a bridge that vibrates too much, the tangling of strands of DNA, and so on. But I have to admit that my own interest is based on the joy of understanding shapes rather than on any particular applications.”

– Burt Totaro, page 64 (a professor I had in college, FWIW)

“As a grad student I had become interested in the annulus conjecture. Saunders Mac Lane advised me that it was a bit hard for a thesis problem (it was), but I thought about it whenever I had an idea. In 1968, while looking after my four-month-old son, an idea occurred to me, now called the “torus trick.” It only took a few days to realize that I had reduced the annulus conjecture to a problem about PL homotopy tori, and in a different direction had proved the local contractibility of the space of homeomorphisms of n-space.”

– Robion Kirby, page 62

“The imprint of the world in our minds is not photographic; all the brain knows of the outside world is a chaotic sequence of electric impulses and out of these it creates a structural entity: our perception of what we see and hear. Most of the time, an adult’s brain talks to itself and creates more and more refined structures within itself. The word “structure” means a mathematical structure, something which becomes more and more abstract and better and better logically organized in the course of this self-conversation….

We are all fascinated with structural patterns: periodicity of a music tune, a symmetry of an ornament, self-similarity of computer images of fractals. And the structures already prepared within ourselves are the most fascinating of all. Alas, most of them are hidden from ourselves…. Brains are our masters, with only 2 percent of our body weight, they take 20 percent of the oxygen resources of our bodies; you cannot cannot resist their commands. You become a mathematician, a slave of this insatiable hunger of your brain, of everybody’s brain, for making structures of everything that goes into it.”

– Mikhael Leonidovich Gromov, page 34

“I often think of cats. I think of trees. I think of dogs occasionally but I don’t think of them all that much because dogs are agreeable. They do what you want them to do to some extent. Some people believe that mathematics is what we think it is and it’s created by our thoughts. I don’t. I’m a Platonist at heart, although I know there are a very great difficulties with that view.”

– John Horton Conway, page 18

“At the moment, one of the things I’m working on understanding is the total wavelength of a surface like a sphere or something of greater complexity, such as the surface of a bagel or a pretzel. What is the total wavelength? … I first became interested in the total wavelength as a model related to a question which can be roughly stated as, can one hear the shape of the universe?”

– Kate Abedola Okikiolu, page 98

“For example, the “Ode to Joy” would be 334554321123322 for the right hand, and 332112345543344 for the left, with corresponding digits always adding up to 6. Soon music became a passion itself, on a par with my passion for numbers, though on its own terms.”

– Noam K Elkies, page 158

“In mathematics, there are not only theorems. There are, what we call, “philosophies” or “yogas,” which remain vague. Sometimes we can guess the flavor of what should be true but cannot make a precise statement. When I want to understand a problem, I first need to have a panorama of what is around it. A philosophy creates a panorama where you can put the things in place and understand that if you can do something here, you can make progress somewhere else. This is how things begin to fit together.”

– Viscount Pierre Deligne, page 156

” I prefer to close my eyes when I think about mathematics. The best work is done by night, in half sleep. Sometimes I go to bed thinking, “Ah, I have a nice lemma to prove–or disprove.” (Should I explain what a lemma is? A mountain climber needs holds to get from one level to the next one. Lemmas are the the holds of a mathematician.)”

– Jean-Pierre Serre, page 144

“How to define the roughness of rusted iron, of broken stone, metal, or glass? What shape is a mountain, a coastline, a river, or a dividing line between two watersheds? That is, can geometry deliver what the word seems to promise, namely, truthful measurements of untamed Earth? How fast does the wind blow during a storm? what shape is a cloud, a flame, or a welding? What is the density of galaxies in the universe? What is the volatility of the prices quoted on financial markets? How to compare and hopefully even measure different writers’ vocabularies?”